Question

Determine the sample size n needed to construct a 99​% confidence interval to estimate the population proportion when p overbar =0.68 and the margin of error equals 5​%.

n=_______ ​(Round up to the nearest​ integer.)

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Determine the sample size n needed to construct a 99​% confidence interval to estimate the population mean for the following margins of error when σ=87. ​a) 25 ​b) 40 ​c) 50

Answer 1

z value at 9The following information is provided,
Significance Level, α = 0.01, Margin of Error, E = 0.05

The provided estimate of proportion p is, p = 0.68
The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.68*(1 - 0.68)*(2.58/0.05)^2
n = 579.37

Therefore, the sample size needed to satisfy the condition n >= 579.37 and it must be an integer number, we conclude that the minimum required sample size is n = 580
Ans : Sample size, n = 580

b)

a)

Solution
The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 25, σ = 87

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 87/25)^2
n = 80.61

Therefore, the sample size needed to satisfy the condition n >= 80.61 and it must be an integer number, we conclude that the minimum required sample size is n = 81
Ans : Sample size, n = 81

b)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 40, σ = 87

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 87/40)^2
n = 31.49

Therefore, the sample size needed to satisfy the condition n >= 31.49 and it must be an integer number, we conclude that the minimum required sample size is n = 32
Ans : Sample size, n = 32

c)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 50, σ = 87

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 87/50)^2
n = 20.15

Therefore, the sample size needed to satisfy the condition n >= 20.15 and it must be an integer number, we conclude that the minimum required sample size is n = 21
Ans : Sample size, n = 21